The Shape of an Image: A Study of Mapper on Images
This work addresses a domain-specific problem in computational topology for image analysis, offering an incremental improvement over existing methods.
The paper tackles the limitation of contour trees in handling real-world images by proposing a customized Mapper construction that assumes only continuity, enabling robust topological analysis for a broader set of images, and provides a fast algorithm and simplification method.
We study the topological construction called Mapper in the context of simply connected domains, in particular on images. The Mapper construction can be considered as a generalization for contour, split, and joint trees on simply connected domains. A contour tree on an image domain assumes the height function to be a piecewise linear Morse function. This is a rather restrictive class of functions and does not allow us to explore the topology for most real world images. The Mapper construction avoids this limitation by assuming only continuity on the height function allowing this construction to robustly deal with a significant larger set of images. We provide a customized construction for Mapper on images, give a fast algorithm to compute it, and show how to simplify the Mapper structure in this case. Finally, we provide a simple procedure that guarantees the equivalence of Mapper to contour, join, and split trees on a simply connected domain.